Distribution Theory/Support and singular support

Definition (being zero on open sets):

Let ${\displaystyle M}$ be a smooth manifold, let ${\displaystyle U\subseteq M}$ be open, and let ${\displaystyle T\in {\mathcal {D}}'(U)}$. Let ${\displaystyle V\subseteq U}$ be an open subset. We say that ${\displaystyle T}$ is zero on ${\displaystyle V}$ iff for all ${\displaystyle \varphi \in {\mathcal {D}}(V)}$ we have ${\displaystyle T(\varphi )=0}$.

Proposition (distribution is zero on union of opens where it is zero):

Let ${\displaystyle U\subseteq M}$ (${\displaystyle M}$ being a smooth manifold), and let ${\displaystyle T\in {\mathcal {D}}'(U)}$. Suppose that ${\displaystyle T}$ is zero on a family of open subsets ${\displaystyle V_{\alpha }\subseteq U}$ (${\displaystyle \alpha \in A}$). Then ${\displaystyle T}$ is also zero on

${\displaystyle V:=\bigcup _{\alpha \in A}V_{\alpha }}$.

Proof: Let ${\displaystyle \varphi \in {\mathcal {D}}(V)}$. Then ${\displaystyle K:=\operatorname {supp} \varphi }$ is a compact subset of ${\displaystyle V}$. Hence, extract a finite subcover ${\displaystyle K\cap V_{\alpha _{1}},\ldots ,K\cap V_{\alpha _{n}}}$. Then pick a finite partition of unity on ${\displaystyle K}$ of functions that are subordinate to ${\displaystyle V_{\alpha _{1}},\ldots ,V_{\alpha _{n}}}$ (using that ${\displaystyle K\cap \bigcup _{i\neq j}(V\setminus V_{\alpha _{i}})}$ is a compact subset of ${\displaystyle V_{\alpha _{j}}}$ and convolving an indicator function on that with a mollifier of sufficiently small support), and use linearity of ${\displaystyle T}$. ${\displaystyle \Box }$

Definition (support):

Let ${\displaystyle T\in {\mathcal {D}}'(U)}$, where ${\displaystyle U}$ is an open subset of a smooth manifold. The support of ${\displaystyle T}$ is the set

${\displaystyle U\setminus \bigcup V}$,

where the union ranges over all open ${\displaystyle V\subseteq U}$ on which ${\displaystyle T}$ is zero.